Misregistration correction

ABSTRACT

A method of misregistration correction in a line scanning imaging system includes: generating a model of scan motion over a focal plane of the imaging system, using a coupled system of scan equations with constant coefficients; estimating programmed motion positions across a plurality of detector junction overlap regions via a state transition matrix solution to the scan equations; at each detector junction overlap region, measuring actual motion positions via image correlation of overlapping detectors; generating differences between the actual motion positions and the estimated programmed motion positions; estimating updates to the constant coefficients based on the generated differences; generating corrections from the estimated updates to remove unwanted motion; and applying the updates to the constant coefficients.

RELATED APPLICATION

This application claims the benefit of U.S. Provisional PatentApplication No. 61/820,902, filed in the U.S. Patent and TrademarkOffice on May 8, 2013, the entire contents of which are incorporatedherein by reference.

BACKGROUND

1. Technical Field

This disclosure relates to generation of images of remote objects and,more particularly, to generation of corrected images generated from ascanning sensor on a moving platform.

2. Discussion of Related Art

Using line scanning, an electro-optical sensor can acquire image datafor a remote object while in motion. Support data associated with eachacquisition by the sensor contains measurement histories for position,velocity, attitude angles and rates, and a priori error variances.Accurate support data is important to the image formation process, aserrors can manifest themselves as geometric artifacts.

Misregistration correction (MRC) is a component in the image formationchain of a line scanning electro-optical sensor. MRC removes thegeometric artifacts introduced by un-programmed motion, which isunwanted motion not commanded as part of the image acquisition process.It can arise from vibrations, sensor attitude and position knowledgeerrors, etc.

SUMMARY

According to one aspect, a method of misregistration correction in aline scanning imaging system is provided. The method includes:generating a model of scan motion over a focal plane of the imagingsystem, using a coupled system of scan equations with constantcoefficients; estimating programmed motion positions across a pluralityof detector junction overlap regions via a state transition matrixsolution to the scan equations; at each detector junction overlapregion, measuring actual motion positions via image correlation ofoverlapping detectors; generating differences between the actual motionpositions and the estimated programmed motion positions; estimatingupdates to the constant coefficients based on the generated differences;generating corrections from the estimated updates to remove unwantedmotion; and applying the updates to the constant coefficients.

In some exemplary embodiments, estimating updates is performed usingleast-squares estimation. In some exemplary embodiments, the imagecorrelation comprises normalized cross-correlation. In some exemplaryembodiments, the image correlation comprises lag productcross-correlation. In some exemplary embodiments, the image correlationcomprises least squares cross-correlation.

In some exemplary embodiments, the model of scan motion is generatedover a predetermined time interval.

In some exemplary embodiments, the scan equations comprise a set ofdifferential equations with constant coefficients. In some exemplaryembodiments, the differential equations are first-order differentialequations.

In some exemplary embodiments, the scan equations are linear.

In some exemplary embodiments, the method further includes computing thescan equation coefficients using sensor platform parameters. In someexemplary embodiments, the sensor platform parameters comprise at leastone of sensor position, velocity, attitude angles and rates. In someexemplary embodiments, the method further includes approximating errorsin the scan equation coefficients from measurement errors in theplatform parameters. In some exemplary embodiments, the method furtherincludes generating a covariance matrix of focal plane coordinateerrors. In some exemplary embodiments, the method further includesestimating platform parameter errors from the differences between theactual motion positions and the estimated programmed motion positions.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure is further described in the detailed descriptionwhich follows, in reference to the noted plurality of drawings by way ofnon-limiting examples of embodiments of the present disclosure, in whichlike reference numerals represent similar parts throughout the severalviews of the drawings.

FIG. 1 includes a schematic diagram of a focal plane scanninggeometry/configuration.

FIG. 2 is a schematic functional flow diagram of a detector junctioncorrelation process.

FIG. 3 includes a schematic functional block diagram of a detectorjunction measurement process.

FIG. 4 is a schematic diagram of traditional pinhole camera projectionequations.

FIG. 5 is a schematic diagram of an error perturbation model, accordingto some exemplary embodiments.

FIG. 6 includes a schematic block diagram of an MRC process, accordingto some exemplary embodiments.

FIG. 7 includes a schematic logical flow diagram of a line-by-line MRCcomputational process, according to some exemplary embodiments.

DETAILED DESCRIPTION

The present disclosure describes an innovative new approach forcomputing and applying required MRC. The approach of the disclosuresimultaneously estimates and corrects un-programmed image motion(linear, oscillatory, and random); while enabling estimation of sensorknowledge errors in position, velocity, attitude angles, and attitudeangle rates. According to the present disclosure, focal plane scanmotion is modeled, over a sufficiently short time interval, via acoupled system of first-order differential equations with constantcoefficients. Programmed motion positions across each detector junctionoverlap region are estimated via a state transition matrix solution tothe scan equations. At each overlap junction, the actual motionpositions are measured via image correlation of the overlappingdetectors, and differences between these and the predicted locations aregenerated. A least-squares estimate is applied to update the scancoefficients based on these error residuals, i.e., differences. Thecorrections necessary to remove the unwanted motion are computed fromthese coefficient updates, and then applied.

Thus, according to some exemplary embodiments, error propagation isformulated via perturbations to the state transition matrix, which doesnot require that motion errors be estimated separately by category(rectilinear, oscillatory, and random). A robust and computationallyefficient algorithm results. The rigorous error propagation modelemployed inherently characterizes statistical consistency, which enablesidentification of junction measurement “outliers,” and reinforcesrobustness. The formulation of scan coefficient updates enablesestimation of knowledge errors in sensor position, velocity, attitudeangles, and attitude angle rates. Accuracy is enhanced by an iterationprocess based on these knowledge errors.

According to some exemplary embodiments, a covariance model relates scancoefficient errors to focal plane coordinate errors. According to theseembodiments, a position in the focal plane of a line scanningelectro-optical sensor is selected at some specified time during a scan.This image point corresponds to some ground point. It is assumed all thesupport data values necessary to define the scan (sensor position andvelocity, attitude angles and rates) and the accuracy with which theseparameters are known (mean errors, covariance), are available. Next, acomputation is made, via closed form expressions, of the errorstatistics (means, covariance) in the projected location of the groundpoint's image at some small time increment later in the scan. Thesestatistics are performance important for the image formation chain.

According to these embodiments, the error propagation occurs in threesteps. First, given random errors (means, and covariance) in the scancoefficients, accurate approximations for the resulting focal planecoordinate errors (means, and covariance) are derived in closed form.Scan coefficient errors arise because the platform parameters (sensorposition, velocity, attitude angles and rates) employed in theircomputation all contain random measurement errors. Next, accurateapproximations for the random errors in the scan coefficients (and theirjoint probability distribution) are derived in closed form from therandom measurement errors in the platform parameters. Finally, combiningthese two steps completes the solution.

According to these embodiments, this new approach produces an accurateclosed-form expression for predicting the performance errors inprojected focal plane scan coordinates. This is a major improvement overthe existing scheme of Monte Carlo simulation, which is computationallycumbersome. Furthermore, the accurate closed form enables on-the-flycomputation of performance quality measures for band-to-band registeredexploitation products. Such metrics are in demand by the user community.These accurate closed-form expressions also facilitate on-the-flystatistical consistency and outlier checks for the junction measurementsthat are important for misregistration correction.

According to some exemplary embodiments, platform error estimation isalso provided. Errors in projected focal plane coordinates are estimatedduring image formation via a correlation process in the detectorjunction overlap regions across the array. A goal is to estimate fromthese measured coordinate residuals knowledge errors in the supportdata; thus, improving image quality.

According to these embodiments, given random errors in the scancoefficients, accurate approximations for the resulting focal planecoordinate errors, as would be measured by the detector junctioncorrelation process, are derived in closed form. Scan coefficient errorsarise because the sensor position, velocity, attitude angles and ratesemployed in their computation contain measurement errors. Expressionsrelating the scan coefficient errors to the measurement errors in theplatform parameters (position, velocity, attitude angles and rates) arethen derived. These two intermediate expressions are then combined intoa least-squares estimator relating measured detector junction coordinateresiduals to platform parameter knowledge errors.

According to these embodiments, this new approach produces accurateestimates for platform parameter knowledge errors from the measureddetector junction coordinate residuals on a line-by-line basis.Improving the accuracy with which platform parameters are knownpositively impacts overall image quality. Furthermore, the errorestimates obtained from the initial execution of this approach can beapplied to the original platform parameter measurement histories, andthe updated histories employed to iterate another pass of junctionmeasurements and further tighten overall knowledge accuracy. Thisapproach facilitates on-the-fly statistical consistency and outlierchecks for the junction measurement process.

MRC approaches employed in previous applications have a number offundamental issues that are rectified by the approach described herein.Previous MRC approaches were formulated for outdated processing speedconstraints, and were replete with heuristics.

A unified approach is described herein that is based on a rigorousmathematical formulation, which is well documented and computationallyrobust. This new MRC approach estimates error propagation through thelinear scan equations via a 1^(st)-order perturbation of the statetransition matrix and the associated forcing vector via additive scancoefficient errors. Consequently, it treats total error without the needfor separately estimating linear, oscillatory and random errors. Therigorous error propagation model employed in the approach describedherein characterizes statistical consistency, which enablesidentification of junction measurement “outliers,” and reinforcesrobustness. Its formulation of scan coefficient updates enablesestimation of knowledge errors in sensor position, velocity, attitudeangles, and attitude angle rates via an inversion process. Accuracy isfurther enhanced by an iteration process based on these knowledgeerrors.

A typical traditional MRC approach is now described as a reference pointfor the detailed description of the approach of the exemplaryembodiments, which begins in section 3 below. In terms of nomenclature,the term “programmed motion” used herein refers to commanded motion thatis planned as part of the sensor's image acquisition process.Specifically, it is the motion of the image of a selected stationaryground point across the focal plane during a commanded scan. The terms“un-programmed motion,” “image motion error” or “focal plane coordinateerrors” refer to the motion of the image of a selected stationary groundpoint across the focal plane, which differs from the commanded motion,and can occur due to attitude errors, vibrations, or a number of othercauses.

Previous electro-optical line scanning systems were based on apost-correction pinhole camera model. Specifically, for these systems itwas assumed that either corrections for optical distortion, atmosphericrefraction, etc., had been successfully executed prior to formulation ofthe focal plane scan equations, or these correction terms wereinsignificant. The resulting linear scan equations were employed toimplement a multi-stage MRC algorithm.

The scan equations were derived by direct differentiation of the pinholecamera projection equations with respect to time. All the parameters ofthe pinhole camera model were allowed to vary with time except for thecoordinates of the ground point of interest, camera focal length, andthe camera principal point coordinates. The resulting scan equationswere a coupled system of linear, 1^(st)-order, differential equationswith six time varying coefficients. Ideally, one would supply the timehistories of the sensor position, velocity, attitude angles, andattitude angle rates, compute the scan six time varying coefficients,and solve the system of differential equations.

In general, these differential equations were found to be a difficult,if not impossible, problem to solve in closed form without thesimplifying assumptions. The simplifying assumption that was adopted isto consider a time increment sufficiently small to ensure that one canaccurately consider the computed scan coefficients to be constant overthis interval. One then solves the resulting equations and utilizes thissolution until the time constraint has been reached. One then repeatsthe process until the entire time period has been coveredincrement-by-increment.

FIG. 1 includes a schematic diagram of a focal plane scanninggeometry/configuration. The scan configuration 10 includes a forwardarray 12 of sensor modules 14 and a trailing array 16 of sensor modules14. Each sensor module 14 includes a plurality of sensors 28 disposedlinearly. The sensor modules 14 of the forward array 12 are staggeredwith respect to the sensor modules 14 of the trailing array 16, suchthat an overlap 20 of sensor modules is created. The forward array 12and trailing array 16 are offset by an offset 18. Through appropriateprocessing of received sensor data, the forward array 12 and trailingarray 16 combine to form a synthetic array 26, as shown in phantom inFIG. 1. During scanning image data collection, an object 24 is firstdetected by the forward array 12, and, due to the motion of the sensorplatform, the object 24 is then detected by trailing array 16. Theobject 24 follows a scan trajectory 22 along the arrays 12 and 16.Previous approaches involved electro-optical line scanning sensors withoffset and overlapping detector placements, as illustrated in FIG. 1.

The solution to the scan equations was employed to predict the objectcrossing points of the leading (forward) array 12 and the trailing array16 of the scan trajectory 22 of an image point. One then correlated thedetector responses of the forward and trailing arrays within eachoverlap region 20 to determine the actual shift in positions of thecrossing points. Any variation between the predicted shift and themeasured shift would be used to estimate the impact of any unplannedmotion present, and correct for it during formation of the syntheticimage array.

One exemplary detector junction correlation process is illustrated inFIG. 2, which is a schematic functional flow diagram of a detectorjunction correlation process. The correlation process employs the factthat the leading and trailing arrays “see” the same image but atdifferent times. The correlator can be a normalized cross correlator, alag product cross correlator, a least squares cross correlator, or otherknown cross correlator. The image correlator is used to determine wherea selected portion of the image viewed by the leading array 12 bestaligns with the image seen by the trailing array 16. This provides ameasured set of focal plane coordinates that can be compared with wherethe match was predicted via the scan equation solution employing supportdata parameters.

In general, the highest measurable spatial frequency of a motiondisturbance is a function of line rate and array separation. FIG. 3includes a schematic functional block diagram of the detector junctionmeasurement process 50. According to the process 50, the differencebetween the measured and predicted focal plane coordinates feed the MRCprocess 52. Referring to FIG. 3, scan coefficients are computed 54,using camera attitude and position data. Using the computed scancoefficients, programmed trajectory is also computed 56. Using thecomputed programmed trajectory, the trailing array crossing point isestimated 58. In parallel, also using camera attitude and position dataas well as the junction overlap data, junction overlaps are correlated60. Using the correlation of the junction overlaps, the trailing arraycrossing point is measured 62. The differences or residuals between thetrailing array crossing estimate 58 and the measured trailing arraycrossing point are computed 64. As noted above, these differences orresiduals are fed to the MRC process 52.

Thus, in previous approaches, the focal plane coordinate error residualsobtained from the detector junction measurements were employed toestimate the amount of un-programmed motion due to unwanted linear,oscillatory, and random motion. The quantified amount of un-programmedmotion in these categories was then “corrected” when forming thesynthetic image array: One subtracted out the unwanted displacementswhen mapping the synthetic array interpolation locations.

In the following sections numbered 1 through 6, preferred embodimentswhich overcome drawbacks and disadvantages of the previous approachesare described in detail.

1. Derivation of the Scan Equations

FIG. 4 is a schematic diagram of traditional pinhole camera projectionequations. Referring to FIG. 4, in some exemplary embodiments, the scanequations are derived by direct differentiation of the pinhole cameraprojection equations in FIG. 4 with respect to time. All the parametersof the pinhole camera model are allowed to vary with time except for thecoordinates of the ground point of interest, camera focal length, andthe camera principal point coordinates.

It is noted that the pinhole projection equations depicted in FIG. 4assume that the position and attitude parameters are referenced to thecentral projection point. It will be understood that, in many systems ofinterest, the attitude measurements are made relative to coordinate axesthat are rotated via fixed amounts relative to the formulation in FIG.4. The fixed angles of this rotation are referred to as interlockangles. In addition, there is a fixed translation relative to thereference point for the position measurements. These effects arerepresented in a modified equation set as follows. The pinhole modelequations in the special case that the perspective center is displacedand the attitude angles must be further rotated by interlock angles aregiven by:

$\begin{matrix}{\begin{pmatrix}{x - x_{p}} \\{y - y_{p}} \\{- f}\end{pmatrix} = {\frac{1}{\lambda}\begin{pmatrix}r_{11} & r_{12} & r_{13} \\r_{21} & r_{22} & r_{23} \\r_{31} & r_{32} & r_{33}\end{pmatrix}\begin{pmatrix}m_{11} & m_{12} & m_{13} \\m_{21} & m_{22} & m_{23} \\m_{31} & m_{32} & m_{33}\end{pmatrix}\begin{pmatrix}\left( {X - X_{c} - X_{0}} \right) \\\left( {Y - Y_{c} - Y_{0}} \right) \\\left( {Z - Z_{c} - Z_{0}} \right)\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 1}\text{-}1}\end{matrix}$

Here, ω, κ, φ represent the traditional roll, pitch, and yaw attitudeangles, respectively. While ω, κ, φ represent the fixed interlockangles. Additionally, X₀,Y₀, Z₀ symbolize the fixed position translationcomponents. Thus, the pinhole model becomes

$\begin{matrix}{\mspace{79mu} {m_{11} = {\cos \; {\varphi cos\kappa}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2a} \\{\mspace{79mu} {m_{12} = {{\cos \; {\omega sin\kappa}} + {\sin \; {\omega sin\varphi cos\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2b} \\{\mspace{79mu} {m_{13} = {{\sin \; {\omega sin\kappa}} - {\cos \; {\omega sin\varphi cos\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2c} \\{\mspace{79mu} {m_{21} = {{- \cos}\; {\varphi sin\kappa}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2d} \\{\mspace{79mu} {m_{22} = {{\cos \; {\omega cos\kappa}} - {\sin \; {\omega sin\varphi sin\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2e} \\{\mspace{79mu} {m_{23} = {{\sin \; {\omega cos\kappa}} + {\cos \; {\omega sin\varphi sin\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2f} \\{\mspace{79mu} {m_{31} = {\sin \; \varphi}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2g} \\{\mspace{79mu} {m_{32} = {{- \sin}\; {\omega cos\varphi}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2h} \\{\mspace{79mu} {m_{33} = {\cos \; {\omega cos\varphi}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2i} \\{\mspace{79mu} {r_{11} = {\cos \; {\varphi cos\varphi\kappa}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2j} \\{\mspace{79mu} {r_{12} = {{\cos \; {\omega sin\kappa}} + {\sin \; {\omega sin\varphi cos\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2k} \\{\mspace{79mu} {r_{13} = {{\sin \; {\omega sin\kappa}} - {\cos \; {\omega sin\varphi cos\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2l} \\{\mspace{79mu} {r_{21} = {{- \cos}\; {\varphi sin\kappa}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2m} \\{\mspace{79mu} {r_{22} = {{\cos \; {\omega cos\kappa}} - {\sin \; {\omega sin\varphi sin\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2n} \\{\mspace{79mu} {r_{23} = {{\sin \; {\omega cos\kappa}} + {\cos \; {\omega sin\varphi sin\kappa}}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2o} \\{\mspace{79mu} {r_{31} = {\sin \; \varphi}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2p} \\{\mspace{79mu} {r_{32} = {{- \sin}\; {\omega cos\varphi}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2q} \\{\mspace{79mu} {r_{33} = {\cos \; {\omega cos\varphi}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2r} \\{\begin{pmatrix}\overset{\sim}{L} \\\overset{\sim}{M} \\\overset{\sim}{N}\end{pmatrix} = {\begin{pmatrix}r_{11} & r_{12} & r_{13} \\r_{21} & r_{22} & r_{23} \\r_{31} & r_{32} & r_{33}\end{pmatrix}\begin{pmatrix}m_{11} & m_{12} & m_{13} \\m_{21} & m_{22} & m_{23} \\m_{31} & m_{32} & m_{33}\end{pmatrix}\begin{pmatrix}\left( {X - X_{c} - X_{0}} \right) \\\left( {Y - Y_{c} - Y_{0}} \right) \\\left( {Z - Z_{c} - Z_{0}} \right)\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 1}\text{-}2s} \\{\mspace{79mu} {{x - x_{p} + {f\frac{\overset{\sim}{L}}{\overset{\sim}{N}}}} = 0}} & {{{Eq}.\mspace{14mu} 1}\text{-}2t} \\{\mspace{85mu} {{y - y_{p} + {f\frac{\overset{\sim}{M}}{\overset{\sim}{N}}}} = 0}} & {{{Eq}.\mspace{14mu} 1}\text{-}2u}\end{matrix}$

Thus, taking the derivative with respect to time of equations 1-2t and1-2u yields

$\begin{matrix}{{\frac{x}{t} + {\left( {f\frac{\overset{\sim}{L}}{\overset{\sim}{N}}} \right)\left\lbrack {{\frac{1}{\overset{\sim}{L}}\frac{\overset{\sim}{L}}{t}} - {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}}} \right\rbrack}} = 0} & {{{Eq}.\mspace{14mu} 1}\text{-}3a} \\{{\frac{y}{t} + {\left( {f\frac{\overset{\sim}{M}}{\overset{\sim}{N}}} \right)\left\lbrack {{\frac{1}{\overset{\sim}{M}}\frac{\overset{\sim}{M}}{t}} - {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}}} \right\rbrack}} = 0} & {{{Eq}.\mspace{14mu} 1}\text{-}3b}\end{matrix}$

Now observe that via equations 1-2t and 1-2u:

$\begin{matrix}{{f\frac{\overset{\sim}{L}}{\overset{\sim}{N}}} = {x_{p} - x}} & {{{Eq}.\mspace{14mu} 1}\text{-}4a} \\{{f\frac{\overset{\sim}{M}}{\overset{\sim}{N}}} = {y_{p} - y}} & {{{Eq}.\mspace{14mu} 1}\text{-}4b}\end{matrix}$

Substituting equation 1-4a into equation 1-3a yields:

$\begin{matrix}{\frac{x}{t} = {{x\left( {\frac{1}{\overset{\sim}{L}}\frac{\overset{\sim}{L}}{t}} \right)} - {x\left( {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}} \right)} - {x_{p}\left\lbrack {{\frac{1}{\overset{\sim}{L}}\frac{\overset{\sim}{L}}{t}} - {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}}} \right\rbrack}}} & {{{Eq}.\mspace{14mu} 1}\text{-}5}\end{matrix}$

Observe that equations 1-2t and 1-2u may be rewritten in the form:

$\begin{matrix}{\frac{x}{\overset{\sim}{L}} = {\frac{y}{\overset{\sim}{M}} + \frac{x_{p}}{\overset{\sim}{L}} - \frac{y_{p}}{\overset{\sim}{M}}}} & {{{Eq}.\mspace{14mu} 1}\text{-}6}\end{matrix}$

Substituting equation 1-6 into the first term on the right hand side of1-5 and regrouping terms one finds:

$\begin{matrix}{\frac{x}{t} = {{x\left\lbrack {\frac{- 1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}} \right\rbrack} + {{y\left( \frac{1}{\overset{\sim}{M}} \right)}\left\lbrack \frac{\overset{\sim}{L}}{t} \right\rbrack} + \left\{ {{x_{p}\left\lbrack {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}} \right\rbrack} - {{y_{p}\left( \frac{1}{\overset{\sim}{M}} \right)}\left\lbrack \frac{\overset{\sim}{L}}{t} \right\rbrack}} \right\}}} & {{{Eq}.\mspace{14mu} 1}\text{-}7}\end{matrix}$

Likewise, substituting 1-4b into 1-3b, and again employing 1-6 yields:

$\begin{matrix}{\frac{y}{t} = {{{- \left( \frac{y}{\overset{\sim}{N}} \right)}\frac{\overset{\sim}{N}}{t}} + {\left( \frac{x}{\overset{\sim}{L}} \right)\frac{\overset{\sim}{M}}{t}} + \left\{ {{y_{p}\left\lbrack {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}} \right\rbrack} - {\frac{x_{p}}{\overset{\sim}{L}}\frac{\overset{\sim}{M}}{t}}} \right\}}} & {{{Eq}.\mspace{14mu} 1}\text{-}8}\end{matrix}$

Thus, the rates of change of the image coordinates across the focalplane (equations 1-7 and 1-8) assume the form of a coupled system offirst order differential equations:

$\begin{matrix}{\frac{x}{t} = {{x\; \Gamma_{1}} + {y\; \Gamma_{2}} + \Gamma_{3}}} & {{{Eq}.\mspace{14mu} 1}\text{-}9a} \\{\frac{y}{t} = {{x\; \Gamma_{4}} + {y\; \Gamma_{5}} + \Gamma_{6}}} & {{{Eq}.\mspace{14mu} 1}\text{-}9b} \\{\Gamma_{1} = {\frac{- 1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}}} & {{{Eq}.\mspace{14mu} 1}\text{-}9c} \\{\Gamma_{2} = {\frac{- 1}{\overset{\sim}{M}}\frac{\overset{\sim}{L}}{t}}} & {{{Eq}.\mspace{14mu} 1}\text{-}9d} \\\begin{matrix}{\Gamma_{3} = \left\{ {{x_{p}\left\lbrack {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}} \right\rbrack} - {{y_{p}\left( \frac{1}{\overset{\sim}{M}} \right)}\left\lbrack \frac{\overset{\sim}{L}}{t} \right\rbrack}} \right\}} \\{= {{{- x_{p}}\Gamma_{1}} - {y_{p}\Gamma_{2}}}}\end{matrix} & {{{Eq}.\mspace{14mu} 1}\text{-}9e} \\{\Gamma_{4} = {\frac{1}{\overset{\sim}{L}}\frac{\overset{\sim}{M}}{t}}} & {{{Eq}.\mspace{14mu} 1}\text{-}9f} \\\begin{matrix}{\Gamma_{5} = \Gamma_{1}} \\{= {\frac{- 1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}}}\end{matrix} & {{{Eq}.\mspace{14mu} 1}\text{-}9g} \\\begin{matrix}{\Gamma_{6} = {{y_{p}\left\lbrack {\frac{1}{\overset{\sim}{N}}\frac{\overset{\sim}{N}}{t}} \right\rbrack} - {\frac{x_{p}}{\overset{\sim}{L}}\frac{\overset{\sim}{M}}{t}}}} \\{= {{{- \Gamma_{5}}y_{p}} - {\Gamma_{4}x_{p}}}}\end{matrix} & {{{Eq}.\mspace{14mu} 1}\text{-}9h}\end{matrix}$

2. An Approximate Solution

This section describes a simplifying assumption that will enable one tosolve equation 1-9 in closed form. This is accomplished by considering atime increment Δt=τ sufficiently small to ensure that one can accuratelyconsider the computed coefficients Γ₁, . . . , Γ₆ to be constant overthis interval. One then solves the resulting equations and utilizes thissolution until the time constraint has been reached. One then repeatsthe process until the entire image capture time has been covered inincrements of Δt=τ.

Observe that equations 1-9a through 1-9h can be written in matrix formas:

$\begin{matrix}{{\frac{\;}{t}{\overset{\rightharpoonup}{s}(t)}} = {{\Lambda_{2 \times 2}{\overset{\rightharpoonup}{s}(t)}} + \overset{\rightharpoonup}{F}}} & {{{Eq}.\mspace{14mu} 2}\text{-}1a} \\{{\overset{\rightharpoonup}{s}(t)} = \begin{pmatrix}{x(t)} \\{y(t)}\end{pmatrix}} & {{{Eq}.\mspace{14mu} 2}\text{-}1b} \\{\Lambda_{2 \times 2} = \begin{pmatrix}\Gamma_{1} & \Gamma_{2} \\\Gamma_{4} & \Gamma_{1}\end{pmatrix}} & {{{Eq}.\mspace{14mu} 2}\text{-}1c} \\{\overset{\rightharpoonup}{F} = \begin{pmatrix}\Gamma_{3} \\\Gamma_{6}\end{pmatrix}} & {{{Eq}.\mspace{14mu} 2}\text{-}1d}\end{matrix}$

Note that the scan coefficients Γ₁, . . . Γ₆ depend on the set ofplatform support parameters.

There are numerous mathematical techniques for solving such a systemincluding: matrix methods, the Laplace transform, method of elimination,etc. Employing matrix methods, one can show that the solution toequation 2-1 is:

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{^{\Lambda {({t - t_{0}})}}{\overset{\rightharpoonup}{s}\left( t_{0} \right)}} + {\int\limits_{t_{0}}^{t}{^{\Lambda {({t - \zeta})}}\overset{\rightharpoonup}{F}{\zeta}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}2}\end{matrix}$

Note that the function e^(A(t-t) ⁰ ⁾ yields a matrix upon evaluation,which in linear systems theory is referred to as the state transitionmatrix. One can represent this function of a matrix in series form as:

$\begin{matrix}{^{\Lambda \; t} = {\sum\limits_{p = 0}^{\infty}\frac{\Lambda^{p}t^{p}}{p!}}} & {{{Eq}.\mspace{14mu} 2}\text{-}3}\end{matrix}$

As the exponential is an entire function in the complex plane, then theseries in equation 2-3 converges for all complex 2×2 matrices.

Evaluation of the exponential function of the matrix Λ may beaccomplished in closed form by employing the spectral resolutiontheorem:

Spectral Resolution Theorem:

If A is a member of the set of n×n complex matrices, ƒ is a functiondefined on the spectrum of A, η_(k) ^((j)) is the value of the j thderivative of ƒ at the eigenvalue λ_(k) (k=1, 2 . . . , s, j=0, . . .m_(k)−1), and m_(k) is the index of λ_(k), then there exists matricesZ_(kj), independent of ƒ, such that:

$\begin{matrix}{{F(A)} = {\sum\limits_{k = 1}^{s}{\sum\limits_{j = 1}^{m_{k}}{f_{k}^{({j - 1})}Z_{kj}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}4}\end{matrix}$

Moreover, the matrices Z_(kj) are linearly independent n×n complexmatrices and commute with A. Recall that the index m_(k) of aneigenvalue λ_(k) of the matrix Λ is the power of the term (λ−λ_(k))^(m)^(k) appearing in the minimal annihilating polynomial of Λ.

Successful application of this theorem requires that one first:

Determine the eigenvalues of the matrix Λ.

Compute the matrices Z_(kj).

The eigenvalues are determined via the characteristic equation:

$\begin{matrix}{{\det {\begin{pmatrix}{\Gamma_{1} - \lambda} & \Gamma_{2} \\\Gamma_{4} & {\Gamma_{1} - \lambda}\end{pmatrix}}} = 0} & {{{Eq}.\mspace{14mu} 2}\text{-}5}\end{matrix}$

The resulting characteristic equation is quadratic:

λ²−2Γ₁λ+(Γ₁ ²−Γ₂Γ₄)=0

One finds the eigenvalues to be:

λ₁=Γ₁+√{square root over (Γ₂Γ₄)}  Eq. 2-6a

λ₂=Γ₁−√{square root over (Γ₂Γ₄)}  Eq. 2-6b

This results in three distinct cases depending on the sign of thediscriminant D:

D=Γ ₂Γ₄  Eq. 2-7

Case 1: D>0 Two distinct real eigenvalues

Case 2: D<0 Two distinct complex conjugate eigenvalues

Case 3: D=0 One repeated real eigenvalue

The spectral resolution of the function e^(Λt) is known for each ofthese cases:Case 1: D>0 Two distinct real eigenvalues

$\begin{matrix}{^{\Lambda {({t - t_{0}})}} = {\frac{^{\alpha {({t - t_{0}})}}}{\beta}\begin{pmatrix}{\beta \; {\cosh \left( {\beta \left( {t - t_{0}} \right)} \right)}} & {\Gamma_{2}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \\{\Gamma_{4}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} & {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 2}\text{-}8a} \\{\alpha = \Gamma_{1}} & {{{Eq}.\mspace{14mu} 2}\text{-}8b} \\{\beta = \sqrt{\Gamma_{2}\Gamma_{4}}} & {{{Eq}.\mspace{14mu} 2}\text{-}8c}\end{matrix}$

Case 2: D<0 Two distinct complex conjugate eigenvalues

$\begin{matrix}{^{\Lambda {({t - t_{0}})}} = {\frac{^{\alpha {({t - t_{0}})}}}{\overset{\sim}{\beta}}\begin{pmatrix}{\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} & {\Gamma_{2}{\sin \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \\{\Gamma_{4}{\sin \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} & {\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 2}\text{-}9a} \\{\overset{\sim}{\beta} = \sqrt{{- \Gamma_{2}}\Gamma_{4}}} & {{{Eq}.\mspace{14mu} 2}\text{-}9b}\end{matrix}$

Case 3: D=0 One repeated real eigenvalue

$\begin{matrix}{^{\Lambda {({t - t_{0}})}} = {^{\alpha {({t - t_{0}})}}\begin{pmatrix}1 & {\Gamma_{2}\left( {t - t_{0}} \right)} \\{\Gamma_{4}\left( {t - t_{0}} \right)} & 1\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 2}\text{-}10}\end{matrix}$

The solution expressed in equation 2-2 may now be computed for eachcase.Case 1: D>0 Two distinct real eigenvalues

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\beta}\begin{pmatrix}{{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} & {\Gamma_{2}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \\{\Gamma_{4}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} & {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)}\end{pmatrix}{\overset{\rightharpoonup}{s}\left( t_{0} \right)}} + {\int_{t_{0}}^{t}{\frac{^{\alpha {({t - ϛ})}}}{\beta}\begin{pmatrix}{{\beta cosh}\left( {\beta \left( {t - ϛ} \right)} \right)} & {\Gamma_{2}{\sinh \left( {\beta \left( {t - ϛ} \right)} \right)}} \\{\Gamma_{4}{\sinh \left( {\beta \left( {t - ϛ} \right)} \right)}} & {{\beta cosh}\left( {\beta \left( {t - ϛ} \right)} \right)}\end{pmatrix}\overset{\rightharpoonup}{F}{\zeta}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}11}\end{matrix}$

Hence, expanding terms

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\beta}\begin{pmatrix}{{\left\lbrack {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} \right\rbrack x_{0}} + {\left\lbrack {\Gamma_{2}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}} \\{{\left\lbrack {\Gamma_{4}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {\beta \; {\cosh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}}\end{pmatrix}} + {\int_{t_{0}}^{t}{\left( \frac{^{\alpha {({t - ϛ})}}}{\beta} \right)\begin{pmatrix}{{\left\lbrack {{\beta cosh}\left( {\beta \left( {t - ϛ} \right)} \right)} \right\rbrack \Gamma_{3}} + {\left\lbrack {\Gamma_{2}{\sinh \left( {\beta \left( {t - ϛ} \right)} \right)}} \right\rbrack \Gamma_{6}}} \\{{\left\lbrack {\Gamma_{4}{\sinh \left( {\beta \left( {t - ϛ} \right)} \right)}} \right\rbrack \Gamma_{3}} + {\left\lbrack {{\beta cosh}\left( {\beta \left( {t - ϛ} \right)} \right)} \right\rbrack \Gamma_{6}}}\end{pmatrix}{ϛ}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}12}\end{matrix}$

Observe that one may rewrite the integral in equation 2-12 to obtain:

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\beta}\begin{pmatrix}{{\left\lbrack {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} \right\rbrack x_{0}} + {\left\lbrack {\Gamma_{2}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}} \\{{\left\lbrack {\Gamma_{4}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} \right\rbrack y_{0}}}\end{pmatrix}} + {\int_{0}^{t - t_{0}}{\left( \frac{^{\alpha \; z}}{\beta} \right)\begin{pmatrix}{\left\lbrack {{\beta\Gamma}_{3}{\cosh \left( {\beta \; z} \right)}} \right\rbrack + \left\lbrack {\left( {\Gamma_{2}\Gamma_{6}} \right){\sinh \left( {\beta \; z} \right)}} \right\rbrack} \\{\left\lbrack {{\beta\Gamma}_{6}{\cosh \left( {\beta \; z} \right)}} \right\rbrack + \left\lbrack {\left( {\Gamma_{4}\Gamma_{3}} \right){\sinh \left( {\beta \; z} \right)}} \right\rbrack}\end{pmatrix}{z}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}13}\end{matrix}$

Direct integration establishes that:

$\begin{matrix}\begin{matrix}{{\int{^{\alpha \; z}{\cosh \left( {\beta \; z} \right)}{z}}} = {{\frac{1}{2}{\int{^{{({\alpha + \beta})}z}{z}}}} + {\frac{1}{2}{\int{^{{({\alpha - \beta})}z}{z}}}}}} \\{= {\frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha + \beta} \right)} + \frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha - \beta} \right)}}}\end{matrix} & {{{Eq}.\mspace{14mu} 2}\text{-}14a} \\\begin{matrix}{{\int{^{\alpha \; z}{\sinh \left( {\beta \; z} \right)}{z}}} = {{\frac{1}{2}{\int{^{{({\alpha + \beta})}z}{z}}}} - {\frac{1}{2}{\int{^{{({\alpha - \beta})}z}{z}}}}}} \\{= {\frac{^{{({\alpha + \beta})}z}}{2\left( {\alpha + \beta} \right)} - \frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha - \beta} \right)}}}\end{matrix} & {{{Eq}.\mspace{14mu} 2}\text{-}14b}\end{matrix}$

Substituting equations 2-14a and 2-14b into 2-13 yields:

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\beta}\begin{pmatrix}{{\left\lbrack {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} \right\rbrack x_{0}} + {\left\lbrack {\Gamma_{2}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}} \\{{\left\lbrack {\Gamma_{4}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} \right\rbrack y_{0}}}\end{pmatrix}} + \begin{pmatrix}{{\Gamma_{3}\begin{bmatrix}{\frac{^{{({\alpha + \beta})}z}}{2\left( {\alpha + \beta} \right)} +} \\\frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}}_{0}^{t - t_{0}} + {\frac{\left( {\Gamma_{2}\Gamma_{6}} \right)}{\beta}\begin{bmatrix}{\frac{^{{({\alpha + \beta})}z}}{2\left( {\alpha + \beta} \right)} -} \\\frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}}_{0}^{t - t_{0}}} \\{{\Gamma_{6}\begin{bmatrix}{\frac{^{{({\alpha + \beta})}z}}{2\left( {\alpha + \beta} \right)} +} \\\frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}}_{0}^{t - t_{0}} + {\frac{\left( {\Gamma_{4}\Gamma_{3}} \right)}{\beta}\begin{bmatrix}{\frac{^{{({\alpha + \beta})}z}}{2\left( {\alpha + \beta} \right)} -} \\\frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}}_{0}^{t - t_{0}}}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 2}\text{-}15}\end{matrix}$

Consequently, the solution to Case 1 (D>0 two distinct real eigenvalues)is:

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\beta}\begin{pmatrix}{{\left\lbrack {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} \right\rbrack x_{0}} + {\left\lbrack {\Gamma_{2}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}} \\{{\left\lbrack {\Gamma_{4}{\sinh \left( {\beta \left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {{\beta cosh}\left( {\beta \left( {t - t_{0}} \right)} \right)} \right\rbrack y_{0}}}\end{pmatrix}} + \begin{pmatrix}{{\Gamma_{3}\begin{bmatrix}{\frac{^{{({\alpha + \beta})}{({t - t_{0}})}}}{2\left( {\alpha + \beta} \right)} +} \\\frac{^{{({\alpha - \beta})}{({t - t_{0}})}}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}} + {\frac{\left( {\Gamma_{2}\Gamma_{6}} \right)}{\beta}\begin{bmatrix}{\frac{^{{({\alpha + \beta})}z}}{2\left( {\alpha + \beta} \right)} -} \\\frac{^{{({\alpha - \beta})}z}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}} - {\frac{\Gamma_{3}}{2}\begin{bmatrix}{\frac{1}{\left( {\alpha + \beta} \right)} +} \\\frac{1}{\left( {\alpha - \beta} \right)}\end{bmatrix}} - {\frac{\left( {\Gamma_{2}\Gamma_{6}} \right)}{2\beta}\begin{bmatrix}{\frac{1}{\left( {\alpha + \beta} \right)} +} \\\frac{1}{\left( {\alpha - \beta} \right)}\end{bmatrix}}} \\{{\Gamma_{6}\begin{bmatrix}{\frac{^{{({\alpha + \beta})}{({t - t_{0}})}}}{2\left( {\alpha + \beta} \right)} +} \\\frac{^{{({\alpha - \beta})}{({t - t_{0}})}}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}} + {\frac{\left( {\Gamma_{4}\Gamma_{3}} \right)}{\beta}\begin{bmatrix}\frac{^{{({\alpha + \beta})}{({t - t_{0}})}}}{2\left( {\alpha + \beta} \right)} \\\frac{^{{({\alpha - \beta})}{({t - t_{0}})}}}{2\left( {\alpha - \beta} \right)}\end{bmatrix}} - {\frac{\Gamma_{6}}{2}\begin{bmatrix}{\frac{1}{\left( {\alpha + \beta} \right)} +} \\\frac{1}{\left( {\alpha - \beta} \right)}\end{bmatrix}} - {\frac{\left( {\Gamma_{4}\Gamma_{3}} \right)}{2\beta}\begin{bmatrix}{\frac{1}{\left( {\alpha + \beta} \right)} +} \\\frac{1}{\left( {\alpha - \beta} \right)}\end{bmatrix}}}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 2}\text{-}16}\end{matrix}$

Case 2: D<0 Two distinct complex conjugate eigenvaluesOne may write immediately via analogy with equation 2-13:

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\overset{\sim}{\beta}}\begin{pmatrix}{{\left\lbrack {\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {\Gamma_{2}{\sin \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}} \\{{\left\lbrack {\Gamma_{4}{\sin \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}}\end{pmatrix}} + {\int_{0}^{t - t_{0}}{\left( \frac{^{\alpha \; z}}{\overset{\sim}{\beta}} \right)\begin{pmatrix}{\left\lbrack {\overset{\sim}{\beta}\Gamma_{3}{\cos\left( {\overset{\sim}{\beta}z} \right)}} \right\rbrack + \left\lbrack {\left( {\Gamma_{2}\Gamma_{6}} \right){\sin\left( {\overset{\sim}{\beta}z} \right)}} \right\rbrack} \\{\left\lbrack {\overset{\sim}{\beta}\Gamma_{6}{\cos\left( {\overset{\sim}{\beta}z} \right)}} \right\rbrack + \left\lbrack {\left( {\Gamma_{4}\Gamma_{3}} \right){\sin\left( {\overset{\sim}{\beta}z} \right)}} \right\rbrack}\end{pmatrix}{z}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}17}\end{matrix}$

One can show that

$\begin{matrix}{{\int{^{\alpha \; z}{\cos\left( {\overset{\sim}{\beta}z} \right)}{z}}} = \frac{^{\alpha \; z}\left( {{{\alpha cos}\left( {\overset{\sim}{\beta}z} \right)} + {\overset{\sim}{\beta}{\sin\left( {\overset{\sim}{\beta}z} \right)}}} \right)}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}}} & {{{Eq}.\mspace{14mu} 2}\text{-}18a} \\{{\int{^{\alpha \; z}{\sin\left( {\overset{\sim}{\beta}z} \right)}{z}}} = \frac{^{\alpha \; z}\left( {{{\alpha sin}\left( {\overset{\sim}{\beta}z} \right)} - {\overset{\sim}{\beta}{\cos\left( {\overset{\sim}{\beta}z} \right)}}} \right)}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}}} & {{{Eq}.\mspace{14mu} 2}\text{-}18b}\end{matrix}$

Substituting equations 2-18a and 2-18b into 2-17 yields:

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\overset{\sim}{\beta}}\begin{pmatrix}{{\left\lbrack {\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {\Gamma_{2}{\sin \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}} \\{{\left\lbrack {\Gamma_{4}{\sin \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack x_{0}} + {\left\lbrack {\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)}} \right\rbrack y_{0}}}\end{pmatrix}} + \left( \begin{matrix}{{\Gamma_{3}\left\lbrack \frac{^{\alpha \; z}\begin{pmatrix}{{{\alpha cos}\left( {\overset{\sim}{\beta}z} \right)} +} \\{\overset{\sim}{\beta}{\sin\left( {\overset{\sim}{\beta}z} \right)}}\end{pmatrix}}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack}_{0}^{t - t_{0}} + {\frac{\left( {\Gamma_{2}\Gamma_{6}} \right)}{\overset{\sim}{\beta}}\left\lbrack \frac{^{\alpha \; z}\begin{pmatrix}{{{\alpha sin}\left( {\overset{\sim}{\beta}z} \right)} -} \\{\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}z} \right)}}\end{pmatrix}}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack}_{0}^{t - t_{0}}} \\{{\Gamma_{6}\left\lbrack \frac{^{\alpha \; z}\begin{pmatrix}{{{\alpha cos}\left( {\overset{\sim}{\beta}z} \right)} +} \\{\overset{\sim}{\beta}{\sin \left( {\overset{\sim}{\beta}z} \right)}}\end{pmatrix}}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack}_{0}^{t - t_{0}} + {\frac{\left( {\Gamma_{4}\Gamma_{3}} \right)}{\overset{\sim}{\beta}}\left\lbrack \frac{^{\alpha \; z}\begin{pmatrix}{{{\alpha sin}\left( {\overset{\sim}{\beta}z} \right)} -} \\{\overset{\sim}{\beta}{\cos \left( {\overset{\sim}{\beta}z} \right)}}\end{pmatrix}}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack}_{0}^{t - t_{0}}}\end{matrix} \right)}} & {{{Eq}.\mspace{14mu} 2}\text{-}19}\end{matrix}$

Hence, one finds the solution for Case 2 (D<0 Two distinct complexconjugate eigenvalues) to be:

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{\frac{^{\alpha {({t - t_{0}})}}}{\overset{\sim}{\beta}}\begin{pmatrix}{{\left\lbrack {\overset{\sim}{\beta}\mspace{11mu} \cos \mspace{11mu} \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)} \right\rbrack x_{0}} + {\left\lbrack {\Gamma_{2}\sin \; \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)} \right\rbrack y_{0}}} \\{{\left\lbrack {\Gamma_{4}\mspace{11mu} \sin \mspace{11mu} \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)} \right\rbrack x_{0}} + {\left\lbrack {\overset{\sim}{\beta}\mspace{11mu} \cos \mspace{11mu} \left( {\overset{\sim}{\beta}\left( {t - t_{0}} \right)} \right)} \right\rbrack y_{0}}}\end{pmatrix}} + \begin{pmatrix}{{\Gamma_{3}\left\lbrack \frac{^{\alpha \; z}\left( {{\alpha \mspace{11mu} \cos \mspace{11mu} \left( {\overset{\sim}{\beta}\; \left( {t - t_{0}} \right)} \right)} + {\overset{\sim}{\beta}\mspace{11mu} \sin \mspace{11mu} \left( {\overset{\sim}{\beta}\left( {t - t_{0}}\; \right)} \right)}} \right)}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack} + {\frac{\left( {\Gamma_{2}\Gamma_{6}} \right)}{\overset{\sim}{\beta}}\left\lbrack \frac{^{\alpha \; z}\left( {{\alpha \mspace{11mu} \sin \mspace{11mu} \left( {\overset{\sim}{\beta}\; \left( {t - t_{0}} \right)} \right)} - {\overset{\sim}{\beta}\mspace{11mu} \cos \mspace{11mu} \left( {\overset{\sim}{\beta}\; \left( {t - t_{0}} \right)} \right)}} \right)}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack} - \frac{\Gamma_{3}\alpha}{\alpha^{2} + {\overset{\sim}{B}}^{2}} + \frac{\left( {\Gamma_{2}\Gamma_{6}} \right)}{\left( {\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right)}} \\{{\Gamma_{6}\left\lbrack \frac{^{\alpha \; z}\left( {{\alpha \mspace{11mu} \cos \mspace{11mu} \left( {\overset{\sim}{\beta}\; \left( {t - t_{0}} \right)} \right)} + {\overset{\sim}{\beta}\mspace{11mu} \sin \mspace{11mu} \left( {\overset{\sim}{\beta}\; \left( {t - t_{0}} \right)} \right)}} \right)}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack} + {\frac{\left( {\Gamma_{4}\Gamma_{3}} \right)}{\overset{\sim}{\beta}}\left\lbrack \frac{^{\alpha \; z}\left( {{\alpha \mspace{11mu} \sin \mspace{11mu} \left( {\overset{\sim}{\beta}\; \left( {t - t_{0}} \right)} \right)} - {\overset{\sim}{\beta}\mspace{11mu} \cos \mspace{11mu} \left( {\overset{\sim}{\beta}\; \left( {t - t_{0}} \right)} \right)}} \right)}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right\rbrack} - \frac{\Gamma_{6}\alpha}{\alpha^{2} + {\overset{\sim}{\beta}}^{2}} + \frac{\left( {\Gamma_{4}\Gamma_{3}} \right)}{\left( {\alpha^{2} + {\overset{\sim}{\beta}}^{2}} \right)}}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 2}\text{-}20}\end{matrix}$

Case 3: D=0 One repeated real eigenvalueSubstituting equation 2-10 into 2-2 yields

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{{^{\alpha {({t - t_{0}})}}\begin{pmatrix}1 & {\Gamma_{2}\left( {t - t_{0}} \right)} \\{\Gamma_{4}\left( {t - t_{0}} \right)} & 1\end{pmatrix}}{\overset{\rightharpoonup}{s}\left( t_{0} \right)}} + {\int_{t_{0}}^{t}{{^{\alpha {({t - ϛ})}}\begin{pmatrix}1 & {\Gamma_{2}\left( {t - ϛ} \right)} \\{\Gamma_{4}\left( {t - ϛ} \right)} & 1\end{pmatrix}}\overset{\rightharpoonup}{F}\ {\zeta}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}21}\end{matrix}$

Hence, one may write that;

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{^{\alpha {({t - t_{0}})}}\begin{pmatrix}{x_{0} + {{\Gamma_{2}\left( {t - t_{0}} \right)}y_{0}}} \\{{{\Gamma_{4}\left( {t - t_{0}} \right)}x_{0}} + y_{0}}\end{pmatrix}} + {\int_{t_{0}}^{t}{{^{\alpha {({t - ϛ})}}\ \begin{pmatrix}{\Gamma_{3} + {\Gamma_{6}{\Gamma_{2}\left( {t - ϛ} \right)}}} \\{{\Gamma_{3}{\Gamma_{4}\left( {t - ϛ} \right)}} + \Gamma_{6}}\end{pmatrix}}{\zeta}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}22}\end{matrix}$

A simple change of variables yields;

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{^{\alpha {({t - t_{0}})}}\begin{pmatrix}{x_{0} + {{\Gamma_{2}\left( {t - t_{0}} \right)}y_{0}}} \\{{{\Gamma_{4}\left( {t - t_{0}} \right)}x_{0}} + y_{0}}\end{pmatrix}} + {\int_{0}^{t - t_{0}}{{^{\alpha \; z}\begin{pmatrix}{\Gamma_{3} + {\Gamma_{6}\Gamma_{2}z}} \\{{\Gamma_{3}\Gamma_{4}z} + \Gamma_{6}}\end{pmatrix}}\ {z}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}23}\end{matrix}$

Consequently,

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{^{\alpha {({t - t_{0}})}}\begin{pmatrix}{x_{0} + {{\Gamma_{2}\left( {t - t_{0}} \right)}y_{0}}} \\{{{\Gamma_{4}\left( {t - t_{0}} \right)}x_{0}} + y_{0}}\end{pmatrix}} + {\int_{0}^{t - t_{0}}{{^{\alpha \; z}\begin{pmatrix}\left( {\Gamma_{3} + {\left\lbrack {\Gamma_{6}\Gamma_{2}} \right\rbrack z}} \right) \\\left( {\Gamma_{6} + {\left\lbrack {\Gamma_{3}\Gamma_{4}} \right\rbrack z}} \right)\end{pmatrix}}\ {z}}}}} & {{{Eq}.\mspace{14mu} 2}\text{-}24}\end{matrix}$

Integrating by parts yields that

$\begin{matrix}{{\int{{^{\alpha \; z}\left\lbrack {\Gamma_{6}\Gamma_{2}} \right\rbrack}z}} = {\left\lbrack {\Gamma_{6}\Gamma_{2}} \right\rbrack \left( \frac{{\alpha \; z} - 1}{\alpha^{2}} \right)^{\alpha \; z}}} & {{{Eq}.\mspace{14mu} 2}\text{-}25a} \\{{\int{{^{\alpha \; z}\left\lbrack {\Gamma_{3}\Gamma_{4}} \right\rbrack}z}} = {\left\lbrack {\Gamma_{3}\Gamma_{4}} \right\rbrack \left( \frac{{\alpha \; z} - 1}{\alpha^{2}} \right)^{\alpha \; z}}} & {{{Eq}.\mspace{14mu} 2}\text{-}25b}\end{matrix}$

Substituting equations 2-25a and 2-25b into 2-24 yields the solution toCase 3 (D=0 One repeated real eigenvalue):

$\begin{matrix}{{\overset{\rightharpoonup}{s}(t)} = {{^{\alpha {({t - t_{0}})}}\begin{pmatrix}{x_{0} + {{\Gamma_{2}\left( {t - t_{0}} \right)}y_{0}}} \\{{{\Gamma_{4}\left( {t - t_{0}} \right)}x_{0}} + y_{0}}\end{pmatrix}} + \begin{pmatrix}{{\left( {\frac{\Gamma_{3}}{\alpha} + {\left\lbrack {\Gamma_{6}\Gamma_{2}} \right\rbrack \left( \frac{{\alpha \left( {t - t_{0}} \right)} - 1}{\alpha^{2}} \right)}} \right)^{\alpha {({t - t_{0}})}}} + \left( {\frac{\Gamma_{3}}{\alpha} + \left( \frac{\left\lbrack {\Gamma_{6}\Gamma_{2}} \right\rbrack}{\alpha^{2}} \right)} \right)} \\{{\left( {\frac{\Gamma_{6}}{\alpha} + {\left\lbrack {\Gamma_{3}\Gamma_{4}} \right\rbrack \left( \frac{{\alpha \left( {t - t_{0}} \right)} - 1}{\alpha^{2}} \right)}} \right)^{\alpha {({t - t_{0}})}}} + \left( {{- \frac{\Gamma_{6}}{\alpha}} + \left( \frac{\left\lbrack {\Gamma_{3}\Gamma_{4}} \right\rbrack}{\alpha^{2}} \right)} \right)}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 2}\text{-}26}\end{matrix}$

One should note that the constant scan coefficients employed in theapproximate solution presented here depend on the coordinates X,Y,Z ofthe ground target point. For each solution one must choose a set ofground coordinates to be used in computing Γ₁, . . . Γ₆. In the casewhere one wishes to specify a point in the focal plane instead; one mayemploy the non-pinhole camera projection equations to establish wherethat point intersects the earth's surface (usually accomplished viaiteration). The resulting ground coordinates may then be employed tocompute the scan coefficients used in the solution.

3. Error Propagation Model and MRC

The solution to the image scan equations is propagated in time via thestate transition matrix and the forcing vector. As the scan coefficientscompletely determine the state transition matrix and forcing vector, anyerrors in image motion propagation arise from errors in the scancoefficients (within the accuracy of the constant scan coefficientsassumption). Thus, the error propagation model can be constructed asindicated in FIG. 5, which is a schematic diagram of an errorperturbation model, according to some exemplary embodiments. FIG. 5illustrates a state at time t₀ and two possible positions at some time tafter t₀. One of the positions is defined by the nominal statetransition matrix and forcing vector provided in FIG. 5, and the otherposition is defined by a perturbed state transition matrix and forcingvector, also provided in FIG. 5. The difference in position between thetwo points is the position error, which is defined by the fundamentalerror propagation equation, also provided in FIG. 5 and reproduced belowas Eq. 3-1a.

Thus the fundamental error propagation equation is:

$\begin{matrix}{{\Delta \; {\overset{\rightharpoonup}{s}(t)}} = {{\left\lbrack {^{{({\Lambda + {\Delta\Lambda}})}{({t - t_{0}})}} - ^{\Lambda {({t - t_{0}})}}} \right\rbrack \overset{\rightharpoonup}{s}\; \left( t_{0} \right)} + {\left\lbrack {{\int_{t_{0}}^{t}{^{{({\Lambda + {\Delta\Lambda}})}{({t - \zeta})}}\left( {\overset{\rightharpoonup}{F} + {\Delta \; \overset{\rightharpoonup}{F}}} \right)}} - {^{\Lambda {({t - \zeta})}}\overset{\rightharpoonup}{F}}} \right\rbrack \ {\zeta}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}1a} \\{\mspace{79mu} {{\Delta \; \Lambda} = \begin{pmatrix}{\Delta \; \Gamma_{1}} & {\Delta \; \Gamma_{2}} \\{\Delta \; \Gamma_{4}} & {\Delta \; \Gamma_{5}}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 3}\text{-}1b} \\{\mspace{79mu} {{\Delta \; \overset{\rightharpoonup}{F}} = \begin{pmatrix}{\Delta\Gamma}_{3} \\{\Delta\Gamma}_{6}\end{pmatrix}}} & {{{Eq}.\mspace{14mu} 3}\text{-}1c}\end{matrix}$

Here Δ{right arrow over (s)}(t) the total error in the focal planecoordinates arising from propagating forward, from the initialconditions {right arrow over (s)}(t₀), in time by (t−t₀) in the presenceof scan coefficient errors ΔΓ₁, . . . ΔΓ₆. It is noted that thisapproach does not require the individual estimation of linear,oscillatory and random errors, as was the case with the previousapproaches.

According to the exemplary embodiments, 1^(st)-order error propagationmodel is constructed by expanding the indicated state transitionmatrices in equation 3-1a in time, and retaining only the first-orderterms:

$\begin{matrix}{^{{({\Lambda + {\Delta\Lambda}})}t} = {{\sum\limits_{p = 0}^{\infty}\; \frac{\left( {\Lambda + {\Delta\Lambda}} \right)^{p}t^{p}}{p!}} \approx {I + {\left( {\Lambda + {\Delta\Lambda}} \right)\left( {t - t_{0}} \right)}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}2a} \\{^{\Lambda {({t - t_{0}})}} = {{\sum\limits_{p = 0}^{\infty}\; \frac{(\Lambda)^{p}\left( {t - t_{0}} \right)^{p}}{p!}} \approx {I + {\Lambda \left( {t - t_{0}} \right)}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}2b}\end{matrix}$

Substituting equations 3-2a and 3-2b into 3-1 and retaining onlyfirst-order terms in the scan coefficient errors ΔΓ₁, . . . , ΔΓ₆yields:

$\begin{matrix}{{\Delta \; {\overset{\rightharpoonup}{s}(t)}} = {{\left\lbrack {{\Delta\Lambda}\left( {t - t_{0}} \right)} \right\rbrack {\overset{\rightharpoonup}{s}\left( t_{0} \right)}} + \left\lbrack {\Delta \; {\overset{\rightharpoonup}{F}\left( {t - t_{0}} \right)}} \right\rbrack + {{\frac{1}{2}\left\lbrack {{({\Delta\Lambda})\overset{\rightharpoonup}{F}} + {\Lambda \left( {\Delta \; \overset{\rightharpoonup}{F}} \right)}} \right\rbrack}\left( {t - t_{0}} \right)^{2}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}3}\end{matrix}$

One observes from equation 3-3 that the focal plane coordinate errorsvary with propagation time. The longer the propagation time, the largerthe resulting focal plane errors will be. This trend has significantpractical impacts for line scanning sensors with slow line rates, orwhere the separation between the leading and trailing detector arrays12, 16 is large.

Thus, according to the disclosure, an equation has been derived thatpredicts focal plane coordinate errors in terms of propagation time andscan coefficient errors. This equation is most useful if one hasknowledge of the scan coefficient errors. The detector junctionmeasurements can be utilized to derive the desired scan coefficientserrors. Specifically, equation 3-3 is rewritten to establish ameasurement sensitivity matrix, which relates the errors in the scancoefficients to the detector-junction-measured coordinate differences(between the predicted and correlation-determined positions).

Equation 3-3 is expanded in terms of the individual scan coefficienterrors ΔΓ₁, . . . , ΔΓ₆, and terms are rearranged to obtain:

$\begin{matrix}{{\Delta \; {\overset{\rightharpoonup}{s}(t)}} = {{H_{2 \times 6}\Delta \; {\overset{\rightharpoonup}{\Gamma}}_{6 \times 1}} = {\begin{pmatrix}H_{11} & H_{12} & H_{13} & H_{14} & H_{15} & H_{16} \\H_{21} & H_{22} & H_{23} & H_{24} & H_{25} & H_{26}\end{pmatrix}\begin{pmatrix}{\Delta\Gamma}_{1} \\{\Delta\Gamma}_{2} \\{\Delta\Gamma}_{3} \\{\Delta\Gamma}_{4} \\{\Delta\Gamma}_{5} \\{\Delta\Gamma}_{6}\end{pmatrix}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4a} \\{\mspace{79mu} {H_{11} = \left\lbrack {{x_{0}\left( {t - t_{0}} \right)} + {\Gamma_{1}\; \frac{\left( {t - t_{0}} \right)^{2}}{2}}} \right\rbrack}} & {{{Eq}.\mspace{14mu} 3}\text{-}4b} \\{\mspace{79mu} {H_{12} = {{y_{0}\left( {t - t_{0}} \right)} + {\frac{\Gamma_{6}}{2}\left( {t - t_{0}} \right)^{2}}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4c} \\{\mspace{79mu} {H_{13} = {\left( {t - t_{0}} \right) + {\frac{\Gamma_{1}}{2}\left( {t - t_{0}} \right)^{2}}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4d} \\{\mspace{79mu} {H_{14} = 0}} & {{{Eq}.\mspace{14mu} 3}\text{-}4e} \\{\mspace{79mu} {H_{15} = 0}} & {{{Eq}.\mspace{14mu} 3}\text{-}4f} \\{\mspace{79mu} {H_{16} = {\Gamma_{2}^{{({t - t_{0}})}^{2}}\text{/}2}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4g} \\{\mspace{79mu} {H_{21} = 0}} & {{{Eq}.\mspace{14mu} 3}\text{-}4h} \\{\mspace{79mu} {H_{22} = 0}} & {{{Eq}.\mspace{14mu} 3}\text{-}4i} \\{\mspace{79mu} {H_{23} = {\frac{\Gamma_{4}}{2}\left( {t - t_{0}} \right)^{2}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4j} \\{\mspace{79mu} {H_{24} = {{x_{0}\left( {t - t_{0}} \right)} + {\Gamma_{3}\; \frac{\left( {t - t_{0}} \right)^{2}}{2}}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4k} \\{\mspace{79mu} {H_{25} = {{y_{0}\left( {t - t_{0}} \right)} + {\Gamma_{6}\; \frac{\left( {t - t_{0}} \right)^{2}}{2}}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4l} \\{\mspace{79mu} {H_{26} = {\left( {t - t_{0}} \right) + {\Gamma_{5}\; \frac{\left( {t - t_{0}} \right)^{2}}{2}}}}} & {{{Eq}.\mspace{14mu} 3}\text{-}4m}\end{matrix}$

At this point, one assumes the scan coefficient errors to be constantacross an image line, and then employs equations 3-4a through 3-4m toexecute a least squares estimate of these scan errors given sufficientdetector junction measurements Δ{right arrow over (s)}_(j)(t₀) j=1 . . .m to over determine the system. It is noted that only the initialcondition {right arrow over (s)}_(j)(t₀) j=1, . . . , m and the scancoefficients Γ_(1j), . . . , Γ_(6j) j=1 . . . , m vary from detectorjunction to junction. Thus the least squares solution becomes:

$\begin{matrix}{{\Delta \; {\overset{\rightharpoonup}{\Gamma}}_{6 \times 1}} = {\left( {H_{6 \times 2m}^{T}H_{2m \times 6}} \right)^{- 1}{H_{6 \times 2m}^{T}\begin{pmatrix}{\Delta \; {{\overset{\rightharpoonup}{s}}_{1}(t)}} \\\vdots \\\vdots \\\vdots \\{\Delta \; {{\overset{\rightharpoonup}{s}}_{m}(t)}}\end{pmatrix}}_{2m \times 1}}} & {{{Eq}.\mspace{14mu} 3}\text{-}5a} \\{{\Delta \; {{\overset{\rightharpoonup}{s}}_{j}(t)}} = \begin{pmatrix}{\Delta \; x_{j}} \\{\Delta \; y_{j}}\end{pmatrix}} & {{{Eq}.\mspace{14mu} 3}\text{-}5b}\end{matrix}$

It is noted that equation 3-5a is only valid when the indicated matrixinversion exists. In the event that the matrix is not invertible for aparticular line, this line would have its total motion error correctionby interpolating the closest lines that yielded successful solutions.

FIG. 6 includes a schematic block diagram of the MRC process describedin detail in the foregoing. Given the estimate of the scan coefficienterrors on a line-by-line basis, the MRC correction process of thedisclosure can be implemented as depicted in FIG. 6. According to theprocess, total motion error is removed prior to synthetic arraygeneration. Referring to FIG. 6, a least squares estimate of Δ{rightarrow over (Γ)}_(6×1) at 140, is computed using camera attitude andposition data and measured junction residuals. Using the least squaresestimate, position errors are computed at 142. Scan coefficients arecomputed at 154, also using the camera attitude and position data. Theprogrammed trajectory is computed at 156 using the computed scancoefficients. The position errors computed at 142 are subtracted at 164from the programmed trajectory computed in 156 to generate the syntheticarray.

4. Covariance Error Model

According to the exemplary embodiments, an error model has beendeveloped that relates a given set of scan coefficient errors to theresulting focal plane coordinate errors. Considering the scancoefficient errors to be random variables with a known joint probabilitydistribution, expressions for the mean and covariance of the resultingfocal plane coordinate errors are obtained. These expressions arereferred to herein as the covariance model.

Define the expectation operation

with respect to the joint probability distribution p(ΔΓ₁, . . . , ΔΓ₆)via:

ƒ(ΔΓ₁, . . . , ΔΓ₆)

=∫∫∫ƒ(ΔΓ₁, . . . , ΔΓ₆)p(ΔΓ₁, . . . , ΔΓ₆)d(ΔΓ₁) . . . d(ΔΓ₆)  Eq. 4-1

This expectation operator will now be employed to compute the desiredmean and covariance of the focal plane coordinate errors. Hence applyingthe expectation operator to equation 3-4a yields the mean error

Δ{right arrow over (s)}(t)

to be:

Δ{right arrow over (s)}(t)

=H _(2×6)

Δ{right arrow over (Γ)}_(6×1)

  Eq. 4-2

Likewise, the covariance P_(Δ{right arrow over (s)}) of the focal planecoordinate errors is computed via:

P _(Δ{right arrow over (s)})≡

(Δ{right arrow over (s)}(t)−

(Δ{right arrow over (s)}(t)

)(Δ{right arrow over (s)}(t)−

Δ{right arrow over (s)}(t)

)^(T)

  Eq. 4-3a

P _(Δ{right arrow over (s)}) =H _(2×6)

Δ{right arrow over (Γ)}_(6×1)Δ{right arrow over (Γ)}^(T) _(1×6)

H ^(T) _(6×2) −H _(2×6)

Δ{right arrow over (Γ)}_(6×1)

Δ{right arrow over (Γ)}^(T) _(1×6)

H ^(T) _(6×2)  Eq. 4-3b

P _(Δ{right arrow over (s)}) =H _(2×6) P _(ΔΓ) H ^(T) _(6×2)  Eq. 4-3c

Here P_(ΔΓ) is the 6×6 covariance matrix of the scan coefficient errors.A representative scan coefficient covariance is computed by employing asufficient number of the line-to-line individual scan coefficient errorvectors from the least squares process via a “moving window” approach.

While the covariance model does not appear directly in the MRC processequations, it is important to being able to execute “on the-fly”statistical consistence checks and detection junction measurementoutlier rejection.

5. Platform Error Inversion Process

The estimated errors in the scan coefficients are related on aline-by-line basis to knowledge errors in the fifteen parameters of theplatform support data (sensor position (3) and velocity (3), attitudeangles (3) and rates (3), ground point location (3)). It is assumed thatthe underlying platform parameter knowledge errors are constant acrossan image line. One then employs all the valid junction measurements fora particular line and then inverts the 1^(st)-order error model via anover determined least squares process as follows.

Specifically, at each detector junction one has:

$\begin{matrix}{{\Delta \; {\overset{\rightharpoonup}{s}(t)}} = {{H_{2 \times 6}\Delta \; {\overset{\rightharpoonup}{\Gamma}}_{6 \times 1}}=={H_{2 \times 6}D_{6 \times 15}\Delta \; {\overset{\rightharpoonup}{\Psi}}_{15 \times 1}}}} & {{{Eq}.\mspace{14mu} 5}\text{-}1a} \\{D_{6 \times 15} = \begin{pmatrix}\frac{\partial\Gamma_{1}}{\partial\omega} & \frac{\partial\Gamma_{1}}{\partial\varphi} & \frac{\partial\Gamma_{1}}{\partial\kappa} & \frac{\partial\Gamma_{1}}{\partial\overset{.}{\omega}} & \frac{\partial\Gamma_{1}}{\partial\overset{.}{\varphi}} & \frac{\partial\Gamma_{1}}{\partial\overset{.}{\kappa}} & \frac{\partial\Gamma_{1}}{\partial X} & \frac{\partial\Gamma_{1}}{\partial Y} & \frac{\partial\Gamma_{1}}{\partial Z} & \frac{\partial\Gamma_{1}}{\partial X_{c}} & \frac{\partial\Gamma_{1}}{\partial Y_{c}} & \frac{\partial\Gamma_{1}}{\partial Z_{c}} & \frac{\partial\Gamma_{1}}{\partial{\overset{.}{X}}_{c}} & \frac{\partial\Gamma_{1}}{\partial{\overset{.}{Y}}_{c}} & \frac{\partial\Gamma_{1}}{\partial{\overset{.}{Z}}_{c}} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\frac{\partial\Gamma_{6}}{\partial\omega} & \frac{\partial\Gamma_{6}}{\partial\varphi} & \frac{\partial\Gamma_{6}}{\partial\kappa} & \frac{\partial\Gamma_{6}}{\partial\overset{.}{\omega}} & \frac{\partial\Gamma_{6}}{\partial\overset{.}{\varphi}} & \frac{\partial\Gamma_{6}}{\partial\overset{.}{\kappa}} & \frac{\partial\Gamma_{6}}{\partial X} & \frac{\partial\Gamma_{6}}{\partial Y} & \frac{\partial\Gamma_{6}}{\partial Z} & \frac{\partial\Gamma_{6}}{\partial X_{c}} & \frac{\partial\Gamma_{6}}{\partial Y_{c}} & \frac{\partial\Gamma_{6}}{\partial Z_{c}} & \frac{\partial\Gamma_{6}}{\partial{\overset{.}{X}}_{c}} & \frac{\partial\Gamma_{6}}{\partial{\overset{.}{Y}}_{c}} & \frac{\partial\Gamma_{6}}{\partial{\overset{.}{Z}}_{c}}\end{pmatrix}} & {{{Eq}.\mspace{14mu} 5}\text{-}1b} \\{{\Delta \; {\overset{\rightharpoonup}{\Psi}}_{15 \times 1}} = \begin{pmatrix}{\Delta\omega} \\\vdots \\\vdots \\\vdots \\{\Delta \; {\overset{.}{Z}}_{c}}\end{pmatrix}_{15 \times 1}} & {{{Eq}.\mspace{14mu} 5}\text{-}1c}\end{matrix}$

One can employ all the valid junction measurement for a particular lineto write:

$\begin{matrix}{\begin{pmatrix}{\Delta \; {\overset{\rightharpoonup}{s}}_{1}} \\{\Delta \; {\overset{\rightharpoonup}{s}}_{2}} \\\vdots \\\vdots \\{\Delta \; {\overset{\rightharpoonup}{s}}_{M}}\end{pmatrix}_{2M \times 1} = {\begin{pmatrix}{{{}_{}^{}{}_{}^{}}D} \\{{{}_{}^{}{}_{}^{}}D} \\\vdots \\\vdots \\{{{}_{}^{}{}_{}^{}}D}\end{pmatrix}_{2M \times 15}\begin{pmatrix}{\Delta\omega} \\\vdots \\\vdots \\\vdots \\{\Delta \; Z_{c}}\end{pmatrix}_{15 \times 1}}} & {{{Eq}.\mspace{14mu} 5}\text{-}2a} \\{{\Delta \; {\overset{\rightharpoonup}{S}}_{2M \times 1}} = {\Theta_{2M \times 15}{\Delta\Psi}_{15 \times 1}}} & {{{Eq}.\mspace{14mu} 5}\text{-}2b}\end{matrix}$

The resulting least squares solution over M valid junctions is:

$\begin{matrix}{{\Delta \; {\overset{\rightharpoonup}{\Psi}}_{15 \times 1}} = {\left\{ {\left( {\Theta^{T}\Theta} \right)^{- 1}\Theta^{T}} \right\}_{15 \times 2M}\Delta \; {\overset{\rightharpoonup}{S}}_{2M \times 1}}} & {{{Eq}.\mspace{14mu} 5}\text{-}3a} \\{\Theta_{2M \times 15} = \begin{pmatrix}{{{}_{}^{}{}_{}^{}}D} \\{{{}_{}^{}{}_{}^{}}D} \\\vdots \\\vdots \\{{{}_{}^{}{}_{}^{}}D}\end{pmatrix}_{2M \times 15}} & {{{Eq}.\mspace{14mu} 5}\text{-}3b}\end{matrix}$

Hence, one can estimate the platform parameter knowledge errors on aline-by-line basis and employ these estimates along with the a prioriknowledge estimates to improve the knowledge of the platform parametersthroughout an image. Indeed, one can iteratively feedback these improvedplatform position errors to iterate the MRC correction process; thusimproving overall algorithm robustness and accuracy.

6. MRC Summary Description

According to the exemplary embodiments, the MRC approach describedherein in detail includes certain assumptions upon which the approach isconstructed. These assumptions are:

Assumption 1: The resulting system of coupled 1^(st)-order differentialequations may be integrated over the associated time increment byholding the scan coefficients constant.Assumption 2: The error propagation characteristics of the linear scanequations can be determined via a 1^(st)-order perturbation of the statetransition matrix and the associated forcing vector via additive scancoefficient errors.Assumption 3: While the scan coefficients obtained from the linear scanequations will vary at each detector junction; it is assumed that asingle set of scan coefficient errors can be obtained across all thedetector junctions via a least squares process that is consistent withthe measured junction coordinate residuals.Assumption 4: The knowledge errors in the platform support parameterscan be assumed constant across an individual scan line.

FIG. 7 includes a schematic logical flow diagram of the line-by-line MRCcomputational process, according to some exemplary embodiments.Referring to FIG. 7, it is noted that the entire process is iterative,line-by-line. Specifically, initial platform support parameters are usedin step 202 to project an initial point on the leading array to thetrailing array at each detector junction across a line. Next, in step204, correlation at each detector junction is performed to establishactual projected point location. Next, at step 206, at each junction, adifference between projected and measured focal plane coordinates iscomputed. Next, at step 208, the coordinate differences computed at 206are used to estimate, for example, by least-squares estimation, the scancoefficient errors for a line. Next, in step 212, the coordinatedifferences computed at 206 are used to estimate, for example, byleast-squares estimation, errors in platform support parameters. Ifanother iteration is required, in step 210, the estimated errors inplatform support parameters are used to correct the errors, and flowreturns to step 202. In contrast, after step 212, if iteration iscomplete, in step 214, the estimated scan errors are used to correct theprojection from each position along the synthetic array to the actualdetector locations. It is noted that the iteration branch depicted inFIG. 7 is optional based on the accuracy of the estimated errors in thesupport parameters. It should also be noted that in some applicationsthe rate at which the platform position, velocity, attitude angles andrates are measured by on-platform equipment may not support aline-by-line implementation. In these cases one would compute newcorrections every m lines, where m is chosen to be consistent with theinterpolation accuracy supported by the time sampling granularity.

Various embodiments of the above-described systems and methods may beimplemented in digital electronic circuitry, in computer hardware,firmware, and/or software. The implementation can be as a computerprogram product (i.e., a computer program tangibly embodied in aninformation carrier). The implementation can, for example, be in amachine-readable storage device and/or in a propagated signal, forexecution by, or to control the operation of, data processing apparatus.The implementation can, for example, be a programmable processor, acomputer, and/or multiple computers.

A computer program can be written in any form of programming language,including compiled and/or interpreted languages, and the computerprogram can be deployed in any form, including as a stand-alone programor as a subroutine, element, and/or other unit suitable for use in acomputing environment. A computer program can be deployed to be executedon one computer or on multiple computers at one site.

Method steps can be performed by one or more programmable processorsand/or controllers executing a computer program to perform functions ofthe invention by operating on input data and generating output. Methodsteps can also be performed by, and an apparatus can be implemented as,special purpose logic circuitry. The circuitry can, for example, be aFPGA (field programmable gate array) and/or an ASIC(application-specific integrated circuit). Modules, subroutines, andsoftware agents can refer to portions of the computer program, theprocessor, the special circuitry, software, and/or hardware, e.g., acontroller such as a microcontroller, that implements thatfunctionality.

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor receives instructions and data from a read-only memory or arandom access memory or both. The essential elements of a computer are aprocessor for executing instructions and one or more memory devices forstoring instructions and data. Generally, a computer can be operativelycoupled to receive data from and/or transfer data to one or more massstorage devices for storing data, e.g., magnetic, magneto-optical disks,or optical disks.

Data transmission and instructions can also occur over a communicationsnetwork. Information carriers suitable for embodying computer programinstructions and data include all forms of non-volatile memory,including by way of example semiconductor memory devices. Theinformation carriers can, for example, be EPROM, EEPROM, flash memorydevices, magnetic disks, internal hard disks, removable disks,magneto-optical disks, CD-ROM, and/or DVD-ROM disks. The processor andthe memory can be supplemented by and/or incorporated in special purposelogic circuitry.

To provide for interaction with a user, the above described techniquescan be implemented on a computer having a display device. The displaydevice can, for example, be a cathode ray tube (CRT) and/or a liquidcrystal display (LCD) monitor. The interaction with a user can, forexample, be a display of information to the user and a keyboard and apointing device, e.g., a mouse or a trackball, by which the user canprovide input to the computer, e.g., interact with a user interfaceelement. Other kinds of devices can be used to provide for interactionwith a user. Other devices can, for example, be feedback provided to theuser in any form of sensory feedback, e.g., visual feedback, auditoryfeedback, or tactile feedback. Input from the user can, for example, bereceived in any form, including acoustic, speech, and/or tactile input.

The above described techniques can be implemented in a distributedcomputing system that includes a back-end component. The back-endcomponent can, for example, be a data server, a middleware component,and/or an application server. The above described techniques can beimplemented in a distributing computing system that includes a front-endcomponent. The front-end component can, for example, be a clientcomputer having a graphical user interface, a Web browser through whicha user can interact with an example implementation, and/or othergraphical user interfaces for a transmitting device. The components ofthe system can be interconnected by any form or medium of digital datacommunication, e.g., a communication network. Examples of communicationnetworks include a local area network (LAN), a wide area network (WAN),the Internet, wired networks, and/or wireless networks.

The system can include clients and servers. A client and a server aregenerally remote from each other and typically interact through acommunication network. The relationship of client and server arises byvirtue of computer programs running on the respective computers andhaving a client-server relationship to each other.

Packet-based networks can include, for example, the Internet, a carrierinternet protocol (IP) network, e.g., local area network (LAN), widearea network (WAN), campus area network (CAN), metropolitan area network(MAN), home area network (HAN), a private IP network, an IP privatebranch exchange (IPBX), a wireless network, e.g., radio access network(RAN), 802.11 network, 802.16 network, general packet radio service(GPRS) network, HiperLAN, and/or other packet-based networks.Circuit-based networks can include, for example, the public switchedtelephone network (PSTN), a private branch exchange (PBX), a wirelessnetwork, e.g., RAN, Bluetooth, code-division multiple access (CDMA)network, time division multiple access (TDMA) network, global system formobile communications (GSM) network, and/or other circuit-basednetworks.

The computing system can also include one or more computing devices. Acomputing device can include, for example, a computer, a computer with abrowser device, a telephone, an IP phone, a mobile device, e.g.,cellular phone, personal digital assistant (PDA) device, laptopcomputer, electronic mail device, and/or other communication devices.The browser device includes, for example, a computer, e.g., desktopcomputer, laptop computer, with a World Wide Web browser, e.g.,Microsoft® Internet Explorer® available from Microsoft Corporation,Mozilla® Firefox available from Mozilla Corporation. The mobilecomputing device includes, for example, a Blackberry®, iPAD®, iPhone® orother smartphone device.

Whereas many alterations and modifications of the disclosure will nodoubt become apparent to a person of ordinary skill in the art afterhaving read the foregoing description, it is to be understood that theparticular embodiments shown and described by way of illustration are inno way intended to be considered limiting. Further, the subject matterhas been described with reference to particular embodiments, butvariations within the spirit and scope of the disclosure will occur tothose skilled in the art. It is noted that the foregoing examples havebeen provided merely for the purpose of explanation and are in no way tobe construed as limiting of the present disclosure.

While the present disclosure has been described with reference toexample embodiments, it is understood that the words that have been usedherein, are words of description and illustration, rather than words oflimitation. Changes may be made, within the purview of the appendedclaims, as presently stated and as amended, without departing from thescope and spirit of the present disclosure in its aspects.

Although the present disclosure has been described herein with referenceto particular means, materials and embodiments, the present disclosureis not intended to be limited to the particulars disclosed herein;rather, the present disclosure extends to all functionally equivalentstructures, methods and uses, such as are within the scope of theappended claims.

1. A method of misregistration correction in a line scanning imagingsystem, comprising: generating a model of scan motion over a focal planeof the imaging system, using a coupled system of scan equations withconstant coefficients; estimating programmed motion positions across aplurality of detector junction overlap regions via a state transitionmatrix solution to the scan equations; at each detector junction overlapregion, measuring actual motion positions via image correlation ofoverlapping detectors; generating differences between the actual motionpositions and the estimated programmed motion positions; estimatingupdates to the constant coefficients based on the generated differences;generating corrections from the estimated updates to remove unwantedmotion; and applying the updates to the constant coefficients.
 2. Themethod of claim 1, wherein estimating updates is performed usingleast-squares estimation.
 3. The method of claim 1, wherein the imagecorrelation comprises normalized cross-correlation.
 4. The method ofclaim 1, wherein the image correlation comprises lag productcross-correlation.
 5. The method of claim 1, wherein the imagecorrelation comprises least squares cross-correlation.
 6. The method ofclaim 1, wherein the model of scan motion is generated over apredetermined time interval.
 7. The method of claim 1, wherein the scanequations comprise a set of differential equations with constantcoefficients.
 8. The method of claim 7, wherein the differentialequations are first-order differential equations.
 9. The method of claim1, wherein the scan equations are linear.
 10. The method of claim 1,further comprising computing the scan equation coefficients using sensorplatform parameters.
 11. The method of claim 10, wherein the sensorplatform parameters comprise at least one of sensor position, velocity,attitude angles and rates.
 12. The method of claim 11, furthercomprising approximating errors in the scan equation coefficients frommeasurement errors in the platform parameters.
 13. The method of claim12, further comprising generating a covariance matrix of focal planecoordinate errors.
 14. The method of claim 11, further comprisingestimating platform parameter errors from the differences between theactual motion positions and the estimated programmed motion positions.